Dirichlet series
| L(s) = 1 | − 8·2-s + 8·3-s + 36·4-s − 6·5-s − 64·6-s − 4·7-s − 120·8-s + 36·9-s + 48·10-s + 11-s + 288·12-s − 2·13-s + 32·14-s − 48·15-s + 330·16-s + 8·17-s − 288·18-s + 4·19-s − 216·20-s − 32·21-s − 8·22-s − 11·23-s − 960·24-s + 25-s + 16·26-s + 120·27-s − 144·28-s + ⋯ |
| L(s) = 1 | − 5.65·2-s + 4.61·3-s + 18·4-s − 2.68·5-s − 26.1·6-s − 1.51·7-s − 42.4·8-s + 12·9-s + 15.1·10-s + 0.301·11-s + 83.1·12-s − 0.554·13-s + 8.55·14-s − 12.3·15-s + 82.5·16-s + 1.94·17-s − 67.8·18-s + 0.917·19-s − 48.2·20-s − 6.98·21-s − 1.70·22-s − 2.29·23-s − 195.·24-s + 1/5·25-s + 3.13·26-s + 23.0·27-s − 27.2·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 17^{8} \cdot 59^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 17^{8} \cdot 59^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{8} \cdot 17^{8} \cdot 59^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.84337\times 10^{13}\) |
| Root analytic conductor: | \(6.93209\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(8\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{8} \cdot 17^{8} \cdot 59^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T )^{8} \) |
| 3 | \( ( 1 - T )^{8} \) | |
| 17 | \( ( 1 - T )^{8} \) | |
| 59 | \( ( 1 + T )^{8} \) | |
| good | 5 | \( 1 + 6 T + 7 p T^{2} + 133 T^{3} + 494 T^{4} + 288 p T^{5} + 4131 T^{6} + 16 p^{4} T^{7} + 24138 T^{8} + 16 p^{5} T^{9} + 4131 p^{2} T^{10} + 288 p^{4} T^{11} + 494 p^{4} T^{12} + 133 p^{5} T^{13} + 7 p^{7} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 7 | \( 1 + 4 T + 36 T^{2} + 127 T^{3} + 645 T^{4} + 1945 T^{5} + 1061 p T^{6} + 19357 T^{7} + 60672 T^{8} + 19357 p T^{9} + 1061 p^{3} T^{10} + 1945 p^{3} T^{11} + 645 p^{4} T^{12} + 127 p^{5} T^{13} + 36 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 - T + 28 T^{2} - 67 T^{3} + 592 T^{4} - 1536 T^{5} + 8997 T^{6} - 24749 T^{7} + 110558 T^{8} - 24749 p T^{9} + 8997 p^{2} T^{10} - 1536 p^{3} T^{11} + 592 p^{4} T^{12} - 67 p^{5} T^{13} + 28 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 + 2 T + 53 T^{2} + 96 T^{3} + 1523 T^{4} + 178 p T^{5} + 29187 T^{6} + 38268 T^{7} + 427896 T^{8} + 38268 p T^{9} + 29187 p^{2} T^{10} + 178 p^{4} T^{11} + 1523 p^{4} T^{12} + 96 p^{5} T^{13} + 53 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 - 4 T + 108 T^{2} - 298 T^{3} + 5251 T^{4} - 10444 T^{5} + 161108 T^{6} - 249190 T^{7} + 3550216 T^{8} - 249190 p T^{9} + 161108 p^{2} T^{10} - 10444 p^{3} T^{11} + 5251 p^{4} T^{12} - 298 p^{5} T^{13} + 108 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 + 11 T + 100 T^{2} + 489 T^{3} + 2444 T^{4} + 10110 T^{5} + 71157 T^{6} + 433905 T^{7} + 2508440 T^{8} + 433905 p T^{9} + 71157 p^{2} T^{10} + 10110 p^{3} T^{11} + 2444 p^{4} T^{12} + 489 p^{5} T^{13} + 100 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 12 T + 179 T^{2} + 1423 T^{3} + 12700 T^{4} + 76480 T^{5} + 525951 T^{6} + 2682170 T^{7} + 16473666 T^{8} + 2682170 p T^{9} + 525951 p^{2} T^{10} + 76480 p^{3} T^{11} + 12700 p^{4} T^{12} + 1423 p^{5} T^{13} + 179 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 9 T + 176 T^{2} + 1501 T^{3} + 16335 T^{4} + 115369 T^{5} + 934532 T^{6} + 5484469 T^{7} + 35127160 T^{8} + 5484469 p T^{9} + 934532 p^{2} T^{10} + 115369 p^{3} T^{11} + 16335 p^{4} T^{12} + 1501 p^{5} T^{13} + 176 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 + 22 T + 391 T^{2} + 5141 T^{3} + 56638 T^{4} + 533086 T^{5} + 4392001 T^{6} + 31785128 T^{7} + 205785776 T^{8} + 31785128 p T^{9} + 4392001 p^{2} T^{10} + 533086 p^{3} T^{11} + 56638 p^{4} T^{12} + 5141 p^{5} T^{13} + 391 p^{6} T^{14} + 22 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + 19 T + 260 T^{2} + 2775 T^{3} + 26784 T^{4} + 221780 T^{5} + 1723117 T^{6} + 12215681 T^{7} + 82121514 T^{8} + 12215681 p T^{9} + 1723117 p^{2} T^{10} + 221780 p^{3} T^{11} + 26784 p^{4} T^{12} + 2775 p^{5} T^{13} + 260 p^{6} T^{14} + 19 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + 5 T + 168 T^{2} + 557 T^{3} + 11432 T^{4} + 20412 T^{5} + 10495 p T^{6} + 228877 T^{7} + 16424272 T^{8} + 228877 p T^{9} + 10495 p^{3} T^{10} + 20412 p^{3} T^{11} + 11432 p^{4} T^{12} + 557 p^{5} T^{13} + 168 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 + 26 T + 492 T^{2} + 6323 T^{3} + 68189 T^{4} + 596113 T^{5} + 4739424 T^{6} + 33725672 T^{7} + 237394796 T^{8} + 33725672 p T^{9} + 4739424 p^{2} T^{10} + 596113 p^{3} T^{11} + 68189 p^{4} T^{12} + 6323 p^{5} T^{13} + 492 p^{6} T^{14} + 26 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 21 T + 455 T^{2} + 6256 T^{3} + 82279 T^{4} + 859472 T^{5} + 8462593 T^{6} + 70691735 T^{7} + 554252848 T^{8} + 70691735 p T^{9} + 8462593 p^{2} T^{10} + 859472 p^{3} T^{11} + 82279 p^{4} T^{12} + 6256 p^{5} T^{13} + 455 p^{6} T^{14} + 21 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 9 T + 300 T^{2} - 1619 T^{3} + 39328 T^{4} - 155036 T^{5} + 3707793 T^{6} - 13147211 T^{7} + 268087570 T^{8} - 13147211 p T^{9} + 3707793 p^{2} T^{10} - 155036 p^{3} T^{11} + 39328 p^{4} T^{12} - 1619 p^{5} T^{13} + 300 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 26 T + 514 T^{2} - 7659 T^{3} + 97111 T^{4} - 1045991 T^{5} + 10331237 T^{6} - 92351405 T^{7} + 781128642 T^{8} - 92351405 p T^{9} + 10331237 p^{2} T^{10} - 1045991 p^{3} T^{11} + 97111 p^{4} T^{12} - 7659 p^{5} T^{13} + 514 p^{6} T^{14} - 26 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 + 14 T + 306 T^{2} + 2819 T^{3} + 43863 T^{4} + 356263 T^{5} + 4613774 T^{6} + 32461330 T^{7} + 363915232 T^{8} + 32461330 p T^{9} + 4613774 p^{2} T^{10} + 356263 p^{3} T^{11} + 43863 p^{4} T^{12} + 2819 p^{5} T^{13} + 306 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 17 T + 546 T^{2} - 7255 T^{3} + 130006 T^{4} - 1408330 T^{5} + 18097581 T^{6} - 161460869 T^{7} + 1624660828 T^{8} - 161460869 p T^{9} + 18097581 p^{2} T^{10} - 1408330 p^{3} T^{11} + 130006 p^{4} T^{12} - 7255 p^{5} T^{13} + 546 p^{6} T^{14} - 17 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 39 T + 710 T^{2} + 7712 T^{3} + 55699 T^{4} + 247027 T^{5} - 7794 p T^{6} - 31339559 T^{7} - 384243014 T^{8} - 31339559 p T^{9} - 7794 p^{3} T^{10} + 247027 p^{3} T^{11} + 55699 p^{4} T^{12} + 7712 p^{5} T^{13} + 710 p^{6} T^{14} + 39 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 11 T + 588 T^{2} + 5647 T^{3} + 156887 T^{4} + 1294249 T^{5} + 24761240 T^{6} + 171771317 T^{7} + 2526281208 T^{8} + 171771317 p T^{9} + 24761240 p^{2} T^{10} + 1294249 p^{3} T^{11} + 156887 p^{4} T^{12} + 5647 p^{5} T^{13} + 588 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 + 469 T^{2} + 917 T^{3} + 105570 T^{4} + 309532 T^{5} + 15627273 T^{6} + 46220456 T^{7} + 1644960196 T^{8} + 46220456 p T^{9} + 15627273 p^{2} T^{10} + 309532 p^{3} T^{11} + 105570 p^{4} T^{12} + 917 p^{5} T^{13} + 469 p^{6} T^{14} + p^{8} T^{16} \) | |
| 97 | \( 1 - 16 T + 605 T^{2} - 8438 T^{3} + 172985 T^{4} - 2098320 T^{5} + 30307115 T^{6} - 314316010 T^{7} + 3553639420 T^{8} - 314316010 p T^{9} + 30307115 p^{2} T^{10} - 2098320 p^{3} T^{11} + 172985 p^{4} T^{12} - 8438 p^{5} T^{13} + 605 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.56706119390399283398347880614, −3.44309902658862931492857642597, −3.40589127629653091573681988355, −3.25791559618794977229702148869, −3.22625949409368919045317052594, −3.22280248212781968533346370892, −3.01286504946387501679495405626, −3.00619557700050044867619478449, −2.92367731739127915980353526211, −2.47028722292611278813100450035, −2.44136413100086568310123709684, −2.35714034266682429016086595919, −2.24844116542332086765954966305, −2.21475176447742710784086389248, −2.12299350922092022455933037248, −2.09453005606232918560906226706, −1.95168671451951869117647989916, −1.55806017101501346671783268287, −1.46232043141746278947850645481, −1.45941446770223001538366032331, −1.44968415272618618777489012211, −1.26708216073879538998614855818, −1.22520379055880673771344961768, −1.19299509030099631226590280295, −1.12789668312220999079435788608, 0, 0, 0, 0, 0, 0, 0, 0, 1.12789668312220999079435788608, 1.19299509030099631226590280295, 1.22520379055880673771344961768, 1.26708216073879538998614855818, 1.44968415272618618777489012211, 1.45941446770223001538366032331, 1.46232043141746278947850645481, 1.55806017101501346671783268287, 1.95168671451951869117647989916, 2.09453005606232918560906226706, 2.12299350922092022455933037248, 2.21475176447742710784086389248, 2.24844116542332086765954966305, 2.35714034266682429016086595919, 2.44136413100086568310123709684, 2.47028722292611278813100450035, 2.92367731739127915980353526211, 3.00619557700050044867619478449, 3.01286504946387501679495405626, 3.22280248212781968533346370892, 3.22625949409368919045317052594, 3.25791559618794977229702148869, 3.40589127629653091573681988355, 3.44309902658862931492857642597, 3.56706119390399283398347880614
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.